On viscoelastic effects in non-Newtonian steady flows through porous media

This paper presents a mathematical model for describing approximately the viscoelastic effects in non-Newtonian steady flows through a porous medium. The rheological behaviour of power law fluids is considered in the Maxwell model of elastic behaviour of the fluids. The equations governing the steady flow through porous media are derived and an analytical solution of these equations in the case of a simple flow system is obtained. The conditions for which the viscoelastic effects may become observable from the pressure distribution measurements are shown and expressed in terms of some dimensionless groups. These have been found to be relevant in the evaluation of viscoelastic effects in the steady flow through porous media.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Sedimentation of particles in shear flows of viscoelastic fluids with fibers

We study the effect of fiber additives on rheology and sedimentation of particle suspensions in a base viscoelastic suspending fluid in the case when the suspension is subjected to shear flow. We found experimentally that fiber additives (3–6 mm in length and 8–12 μm in diameter at a mass fraction of 0–0.4%) increase the suspension viscosity and retard the particle sedimentation significantly. At the same mass concentration, long and thin fibers reduce the sedimentation velocity and increase the viscosity to a much greater extent than short and thick fibers. We revealed that both rheology and sedimentation are controlled by a single conformational parameter (overlap parameter) defined as the number of fibers per unit volume multiplied by fiber length cubed.     

Sedimentation of particles in shear flows of viscoelastic fluids

An experimental investigation of sedimentation of single particles and concentrated suspensions in Couette flows of viscous and viscoelastic fluids is presented. With the passage from viscous to viscoelastic fluids, a slowing down of the sedimentation was observed. The sedimentation of suspensions in viscoelastic fluids accelerated with increase in the suspension concentration. The development of instabilities during the sedimentation was detected.     

Effects of the side walls on starting flows in ducts

This article considers the effects of the side walls on the unsteady flow of an incompressible viscous fluid in a duct of uniform cross-section. In order to show the effects of the side walls, three illustrative examples are given. They are: the starting flow in a duct of semicircular cross-section, the starting flow in a duct of rectangular cross-section and the starting flow in a duct of circular cross-section. The velocity distributions and the volume fluxes obtained for these flows are compared and it is shown that the flow in a duct of semicircular cross-section reaches steady state earlier than those for the flow in a duct of circular cross-section and for the flow in a duct of square cross-section. It is found that there are remarkable effects of the side walls of a duct on the required time to attain the asymptotic values of flow properties.     

Multi-layer channel flows with yield stress fluids

We present results of a computational study of visco-plastically lubricated plane channel multi-layer flows, in which the yield stress fluid layers are unyielded at the interface. We demonstrate that symmetric 3-layer flows may be established for wide ranges of viscosity ratio (m), Bingham number (B) and interface position (y_i), for Reynolds numbers Re ≤ 100. Here an inner Newtonian layer is sandwiched between 2 layers of Bingham fluid. Results are presented illustrating the variation of development length with the main dimensionless parameters and for different inlet sizes. We also show that these flows may be initiated by injecting either fluid into a steady flow of the other fluid. The flows are established quicker when the core fluid is injected into a channel already full of the outer fluid. In situations where the inner fluid flow rate is dominant we observed inertial symmetry breaking in the symmetric start-up flows as Re was increased. Asymmetry is also observed in studying temporal nonlinear stability of these flows, which appear stable up to moderate Re and significant amplitudes. In general the flows destabilize at lower Re and perturbation amplitudes than do the analogous core-annular pipe flows, but 1–1 comparison is hard. When the flow is stable the decay characteristics are very similar to those of the pipe flows. In the final part of the paper we explore more exotic flow effects. We show how flow control could be used to position layers asymmetrically within the flow, and how this effect might be varied transiently. We demonstrate that more complex layered flows can be stably achieved, e.g. a 7-layered flow is established. We also show how a varying inlet position can be used to “write” in the yield stress fluid: complex structures that are advected with the flow and encapsulated within the unyielded fluid.     

On the flow resistance of viscoelastic fluids through packed beds

Analytical solutions are obtained for the free surface cell model of packed beds using a third order fluid. Second order perturbed results indicate a substantial increase in resistance to the flow of a viscoelastic fluid through a packed bed. This predicted increase is in good agreement with experimental findings.     

Transonic flows around an airfoil in wind tunnels with porous walls

A study is made of two-dimensional transonic flows of gas around an airfoil in the working part of a wind tunnel with porous walls. The values of the flow parameters are determined by the numerical solution of a boundary-value problem for the equation of the velocity potential; this problem simulates the gas flow around the profile in the tunnel with porous walls. The obtained results are then used to construct an asymptotic theory of the influence of the wind-tunnel height and the Mach number M of the flow in it on the characteristics of the flow around the airfoil.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1980.     

Turbulent heat transfer in circular tube flows of viscoelastic fluids

The effects of thermal entrance length, polymer degradation and solvent chemistry were found to be critically important in the determination of the drag and heat transfer behavior of viscoelastic fluids in turbulent pipe flow. The minimum heat transfer asymptotic values in the thermally developing and in the fully developed regions were experimentally determined for relatively high concentration solutions of heat transfer resulting in the following correlations: $$\begin{gathered} j_H = 0.13\left( {\frac{x}{d}} \right)^{ - 0.24} \operatorname{Re} _a^{ - 0.45} thermally developing region \hfill \\ x/d< 450 \hfill \\ j_H = 0.03 \operatorname{Re} _a^{ - 0.45} thermally developed region \hfill \\ x/d< 450 \hfill \\ \end{gathered} $$ For dilute polymer solutions the heat transfer is a function ofx/d, the Reynolds number and the polymer concentration. The Reynolds analogy between momentum and heat transfer which has been widely used in the literature for Newtonian fluids is found not to apply in the case of drag-reducing viscoelastic fluids.     

Numerical simulation for rotating internal weakly viscoelastic flows in rectangular ducts

The present work develops a numerical method for the solution of rotating internal weakly viscoelastic flows in rectangular ducts for dimensionless parameters such as the Reynolds, Rossby and Weissenberg numbers, taken respectively in the intervals between 171 and 12000, 0.047 and 1/12 and up to 1/10000. It is shown that the usual counter‐rotating double‐vortex configuration of secondary flow breaks down with the increase of the Reynolds number (over the threshold of 171). For higher Reynolds numbers such as 7500 and 12000 the secondary flow diffuses to the interior of the duct where it assumes a fully developed configuration and the transition to the turbulence structure is observed. The Sobolev norms increase almost proportionally to the increase of the Reynolds number, and play an essential role for more complex problems involving transition to turbulence modelling. Copyright © 2002 John Wiley & Sons, Ltd.     

Small perturbation spectrum for plane-parallel flows of viscoelastic fluids

The investigation of the occurrence of a transition from the laminar to the turbulent flow regime in weak polymer solutions is of great practical interest. Experimental data indicate both an increase in flow stability and an occurrence of early turbulence [1]. Paper [2] explains the discrepancy in the experimental data for the numerical investigation of the first-mode symmetric perturbations, which are unstable for a Newtonian fluid. Paper [3] shows that other modes also become unstable in the case of the flow of a viscoelastic Maxwellian fluid in a channel. These features of the hydrodynamic stability of viscoelastic fluids indicate a significant rearrangement of the small perturbation spectrum. In the present paper, the perturbation spectrum for plane-parallel flows of viscoelastic Oldroyd and Maxwellian fluids is investigated at small Reynolds numbers, and at large and small wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–167, November–December, 1984.     

Anomalous transport phenomena in rapid external flows of viscoelastic fluids

Summary The concept of an elastic boundary layer is proposed to explain certain anomalous transport phenomena which occur during rapid external flows of viscoelastic fluids past immersed objects. Reported experimental observations are interpreted by using models based on this concept. Particularly, data on velocity independent drag and heat transfer coefficients for flow of dilute polymer solutions past tiny cylinders are satisfactorily correlated.

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Zusammenfassung Es wird das Konzept einer elastischen Grenzschicht entworfen, um gewisse anomale Transportphänomene zu erklären, welche bei schnellen Strömungen viskoelastischer Flüssigkeiten um eingetauchte Körper auftreten. Die berichteten experimentellen Beobachtungen werden mit Hilfe von Modellen interpretiert, die auf diesem Konzept basieren. Insbesondere werden Daten über geschwindigkeitsunabhängige Widerstands- und Wärmeübertragungs-Koeffizienten bei der Strömung verdünnter Polymerlösungen um dünne Zylinder befriedigend korreliert.

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A, B numerical constants - A ₁,A ₂ surface areas - C _D drag coefficient - D cylinder diameter - F hoop force - h heat transfer coefficient - k thermal conductivity - M molecular weight - Nu Nusselt number - R gas constant - T absolute temperature - u x-component of the velocity - U free stream velocity - x, y Cartesian coordinates - shear rate - boundary layer thickness - ₀ elastic boundary layer thickness - relaxation time - µ viscosity - v kinematic viscosity - [] intrinsic viscosity - density - normal stress difference - shear stress With 3 figures     

Fully-developed pipe and planar flows of multimode viscoelastic fluids

Two solutions are presented for fully-developed pipe and planar flows of multimode viscoelastic models. The fluids have a Newtonian solvent contribution and the polymer modes are described by the Phan-Thien—Tanner (PTT), the FENE-P or the Giesekus equation. The first solution is exact and can handle any number of modes, but is only semi-analytical. The second solution, which is presented only for the PTT model with a linear stress coefficient and the FENE-P model, can also handle any number of modes. It is based on a truncated series expansion and is completely analytical, but provides only an approximated solution. The complexity of the multimode solutions is investigated first with the exact semi-analytical method and it is shown that at high Deborah number flows the high-order stresses can become as important as the stress of the first mode. It is also under these conditions that the approximated analytical solution deviates from the exact semi-analytical solution. A criterion for the accurate use of the approximated solution is presented. Fortran codes are provided to obtain these solutions at the internet address at the end.     

The normal stress behaviour of suspensions with viscoelastic matrix fluids

 We investigate the variations in the shear stress and the first and second normal stress differences of suspensions formulated with viscoelastic fluids as the suspending medium. The test materials comprise two different silicone oils for the matrix fluids and glass spheres of two different mean diameters spanning a range of volume fractions between 5 and 25%. In agreement with previous investigations, the shear stress–shear rate functions of the viscoelastic suspensions were found to be of the same form as the viscometric functions of their matrix fluids, but progressively shifted along the shear rate axis to lower shear rates with increasing solid fraction. The normal stress differences in all of the suspensions examined can be conveniently represented as functions of the shear stress in the fluid. When plotted in this form, the first normal stress difference, as measured with a cone and plate rheometer, is positive in magnitude but strongly decreases with increasing solid fraction. The contributions of the first and the second normal stress differences are separated by using normal force measurements with parallel plate fixtures in conjunction with the cone-and-plate observations. In this way it is possible for the first time to quantify successfully the variations in the second normal stress difference of viscoelastic suspensions for solid fractions of up to 25 vol.%. In contrast to measurements of the first normal stress difference, the second normal stress difference is negative with a magnitude that increases with increasing solid content. The changes in the first and second normal stress differences are also strongly correlated to each other: The relative increase in the second normal stress difference is equal to the relative decrease of the first normal stress difference at the same solid fraction. The variations of the first as well as of the second normal stress difference are represented by power law functions of the shear stress with an unique power law exponent that is independent of the solid fraction. The well known edge effects that arise in cone-and-plate as well as parallel-plate rheometry and limit the accessible measuring range in highly viscoelastic materials to low shear rates could be partially suppressed by utilizing a custom- designed guard-ring arrangement. A procedure to correct the guard-ring influence on torque and normal force measurements is also presented. Received: 20 December 2000 Accepted: 7 May 2001     

An anomalous phenomenon occurring in the flow of viscoelastic fluids out of ducts

When viscoelastic fluids flow out from horizontal ducts many cracks and protruding ridges are formed on the jet surface. As far as the authors know, this phenomenon has not yet been reported. The occurrence of the anomalous phenomenon is not affected by inlet flow condition or duct shape.The phenomenon may be divided into three regimes, namely, a stable state, a breakage state, and a multiplication state. In the breakage state one ridge divides into two parts after its growth, and in the multiplication state a new ridge is suddenly formed at a crack point. With increasing shear rate, the flow pattern of the jet changes from the stable state to the breakage state, and then to the multiplication state.Furthermore, a recoil effect is observed. In this effect many air bubbles rush into the duct from the exit when the flow is quickly stopped.